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Theorem annotanannot 863
Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.)
Assertion
Ref Expression
annotanannot ((𝜑 ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem annotanannot
StepHypRef Expression
1 ibar 524 . . . 4 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
21bicomd 214 . . 3 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
32notbid 309 . 2 (𝜑 → (¬ (𝜑𝜓) ↔ ¬ 𝜓))
43pm5.32i 570 1 ((𝜑 ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 198  df-an 385
This theorem is referenced by:  clwwlknclwwlkdif  27207
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